Abstract: In real life, people are not completely rational. Then when two competitors make decisions from their own selfish point of view at the same time, it is not always the best to maximize their own interests or minimize their own costs. On the other hand, there exist much uncertainty and incomplete information in a game. Based on the fact that there exists uncertainty in non-cooperative games or it is not always appropriate from one's own selfish perspective, we assume that competitors make decisions fromthe view of the lowest cost of the opponent and uncertainty. Then a robust cooperative dual equilibria model with strategy uncertainty for two players is introduced. In the model, each player wants to minimize the opponent's cost where he/she knows his/her cost matrix exactly while he/she can not evaluate his/her own strategy set accurately. Furthermore, the strategy set of each player can be estimated at a symmetric bounded closed set which is a subset of the mixed strategy set. In what follows, the robust optimization technology and dual theory are adopted. Then some results are obtained as follows:
When elements in his/her own uncertain strategy set are taken as l₂-norm, the problem of the lowest cost of the opponent can be transformed to a second-order cone optimization problem, while the problem of the lowest costs of both sides can be converted to a second-order cone complementarity problem. When the element in the uncertain strategy set is considered as l_1∩∞-norm, the corresponding problem of the lowest cost of the opponent can be transformed to a linear programming problem, and the problem of the lowest costs of both sides can be converted to a mixed complementarity problem. Finally, we present a numerical experiment to illustrate the reasonability and validity of the model. Furthermore, weshow that the model can be applied to the optimal reinsurance.
In the work, strategy uncertainty and cooperation are discussed at the same time. It can be regarded as a generalization and supplement of a non-cooperative bimatrix game.

Key words: robust cooperative dual equilibria, symmetric uncertainty strategy set, l₂-norm, l_1∩∞-norm, second order cone complementarity problem, mixed complementarity problem

摘要： 文章从竞争对手角度出发,提出合作对偶均衡:博弈双方自身支付矩阵能准确获知,其自身策略集不能准确获知,但可估计其策略集落在一有界对称闭集且是混合策略集子集内,双方同时做出决策,使得对手成本最低。接着采用鲁棒优化技术和对偶理论进行研究,得到:当自身不确定策略集中元素取l₂-范数时,双方成本同时最低的问题可转化成一个二阶锥互补问题。当不确定策略集中元素取l_1∩∞-范数时,双方成本同时最低问题可转化成一个混合互补问题。最后选取一个数值算例,对模型的合理性和有效性进行验证。

关键词: 鲁棒合作对偶均衡, 对称不确定策略集, l₂-范数, l_1∩∞-范数, 二阶锥互补问题, 混合互补问题